In conventional rotary equipment variations and compliances in gear drive systems coupled with process parameters such as speed, stroke rate, punch frequency , motor commutator frequency, etc., cause non-uniform motion transmission and structural vibration and noise. These effects are well known in rotary slitters, rotary punchers, choppers, web conveyance machines, copiers, printers and the like, in which motion transmission between axes of rotation is achieved through a set of mating spur or helical gears. In applications that require small axial clearances (e.g. between a punch and a die), well-designed spur gears are favored over helical gears as they show lower vibration in the axial direction. Spur gears have no significant capability to impart axial errors. However, there is an enormous need to design and manufacture high quality spur gears that transmit motion uniformly and cause lower structural vibration and noise in rotary equipment. Thus, the design and manufacture of high quality spur gears have been a challenge for gear engineers for quite some time.
According to FIGS. 1a , 1b, 1c, a typical spur gear (partial representation illustrated) has a plurality of teeth 3 arranged 360 degrees about gear body 4. There are several important features of spur gears which are defined herein. One important feature of gear 1 is diametral pitch (P.sub.d) Diametral pitch (P.sub.d) is a measure of the size of the gear tooth 3. The higher the value of diametral pitch (P.sub.d), the smaller is the size of the gear tooth. More particularly, the diametral pitch (P.sub.d) is the ratio of the number of teeth in the gear to its standard pitch diameter.
Addendum, another important feature of spur gears, is the radial distance between the tip circle 5 and the standard pitch circle 6. Profile angle (.phi.) is defined at the point 7 of intersection of the standard pitch circle 6 and the tooth profile. Profile angle (.phi.) is equal to the angle between a line normal 8 to the pitch circle 6 and a line tangent 9 to the tooth profile. Addendum modification (effects illustrated in FIG. 1 (b)) is the amount by which a gear is enlarged or reduced in size generally to satisfy one or more particular constraints.
The procedure for designing gears, particularly spur gears, generally starts with the physical space requirements such as center distance, speed ratio between output and input shafts, etc. As shown in FIG. 2, a flow chart depicts the existing method commonly used in the art for designing spur gears. Broadly described, the current method of designing spur gears involves a combination of two steps, namely synthesis 10 and analysis 16. According to FIG. 2, the synthesis step 10 generates a set of feasible solutions that satisfy physical space constraints or requirements which the gears must satisfy. Thus, in the synthesis step 10, for each feasible solution the geometric design variables 12, such as, number of teeth, profile angle, diametral pitch, and face width, are specified.
In the analysis step 16 (FIG. 2),candidate gear designs 14 are evaluated for performance requirements such as transmitted load/torque needs, vibration, noise, reliability, lubrication, life, etc. Additionally, gear tooth profiles are modified to eliminate interferences that can result from gear tooth and shaft deflections under load and errors in manufacturing and assembly. These modifications, also known as profile modifications (FIG. 1c), help to minimize non-uniform motion, vibration, and noise in load carrying spur gear meshes. If the analysis does not yield acceptable gear designs based on requirements imposed in the synthesis step 10, spur gears having other design parameters 12 are designed and then subjected to analysis 16 in the same fashion.
Skilled artisans will appreciate that selection of number of gear teeth 3 (FIG. 1) has been primarily based on speed ratio, strength, and weight requirements. Additionally, it is well known that the number of teeth 3 in a spur gear should be an integer. This requirement, experience has taught, constrains the actual speed ratio that a given pair of gears can transmit. For a given profile angle (most commonly used is 20 degrees), spur gears with less than 24 teeth usually require addendum modification (profile shift) to avoid undercut or undesirable contact near the base circle of the gear.
In some rotary process applications, the number of gear teeth 3 are chosen such that the numbers are relatively prime. This minimizes the number of times the same tooth pair in the mating gears contacts each other. Erichello discloses a procedure presented in "A Rational Procedure For Designing Minimum Weight Gears," Proceedings of the 1989 ASME International Power Transmission and Gearing Conference, Chicago, Vol. 1, pp. 111-114, for designing minimum weight spur gears and emphasized the presence of an optimum number of teeth in the pinion (mating gear with the smaller number of teeth) that maximizes the load capacity of the gear set.
Presently, a method of making spur gears has not been disclosed in the prior art that involves the selection of number of teeth based on either gear manufacturing requirements or noise and vibration requirements of the rotary equipment.
Those skilled in the art will also appreciate the importance of contact ratio of spur gear meshes (not shown) to overall gear performance. The contact ratio in spur gear meshes is a measure of average number of tooth pairs in contact. For a spur gear mesh with contact ratio equal to 1.6, two tooth pairs are in contact 60% of the time and one tooth pair is in contact 40% of the time. Contact ratio in a given spur gear mesh depends upon variables such as number of teeth, pressure angle, center distance, and diametral pitch. Spur gear meshes with a contact ratio less than 2.0 are called Low Contact Ratio (LCR) gear meshes. Spur gears with contact ratio greater than 2.0 are called High Contact Ratio (HCR) gear meshes. In HCR spur gear meshes, the number of tooth pairs in contact vary from 2 to 3. If an HCR gear mesh has a contact ratio of 2.2, then 3 tooth pairs are in contact 20% of the time and 2 tooth pairs are in contact 80% of the time. As the mating gears rotate, stiffness of the gear changes considerably when number of tooth pairs in contact change from 3 to 2 (in HCR meshes) or from 2 to 1 (in LCR meshes). This periodic change in mesh stiffness results in periodic variation in tooth deflections under load. These tooth deflections coupled with errors in tooth profiles result in periodic variation in motion transmission between the drive and the driven gears. This variation in motion transmission is widely known in the gearing industry as transmission error. In addition, at higher speeds, the transmission error and variation in mesh stiffness act as a dynamic excitor for gear mesh vibration and noise. As the percentage change in mesh stiffness for HCR meshes (change from 2 tooth pairs to 3 tooth pairs) is lower than the percent change in mesh stiffness for LCR meshes (change from 1 tooth pair to 2 tooth pairs), one can expect high quality HCR gear meshes to have lower mesh induced vibration and noise than LCR gear meshes.
It is well known that HCR gear meshes have higher load capacity and show lower gear mesh induced vibration and noise than LCR gear meshes. (See for instance "How to Design Quiet Transmissions," MACHINE DESIGN, by Raymond J. Drago, Dec. 1980; and "Gear Design," SAE Paper No. 680381, by John C. Leming).
Moreover, U.S. Pat. No. 1,525,642 by Anthony B. Cox, discloses gears with integral contact ratios that exhibit constant number of tooth pairs in contact. Such gears have constant length of line of theoretical tooth contact carrying the transmitted load.
Further, the use of integer contact ratios to achieve constant mesh stiffness which show substantial noise reduction over a range of loads is disclosed in "The Reduction of Gear Pair Transmission Error by minimizing Mesh Stiffness Variation," American Gear Manufacturers Association, Paper No. 88-FTM-11, by W. S. Rouverol and W. J. Pearce. In order to achieve constant mesh stiffness, one needs to balance the compliances of the pinion and the gear tooth. Balancing the compliances of the pinion and the gear is achieved by varying the thicknesses of the tooth. However, the benefit of integral contact ratios is lost when errors in tooth profile during manufacturing change the actual contact ratio to a non-integral value.
Current gear design procedures vary geometric gear design variables to achieve designs with high contact ratio that meet strength and space requirements. To date, no studies have been reported on the interaction between selection of contact ratio and operating speeds of the rotary equipment during the gear design process.
Therefore, a need persists in the art for a method for designing a uniform motion, substantially vibration-free spur gears useful in rotary processes that take into account such factors as contact ratio and number of gear teeth, and that is reliable, easy to use and requires a relatively short production cycle time to carry out.